Introduction to turbulence/Statistical analysis/Estimation from a finite number of realizations
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X_{N}=\frac{1}{N}\sum^{N}_{n=1} x_{n} | X_{N}=\frac{1}{N}\sum^{N}_{n=1} x_{n} | ||
+ | </math> | ||
+ | </td><td width="5%">(2)</td></tr></table> | ||
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+ | Now, as we observed in our simple coin-flipping experiment, since the <math>x_{n}</math> are random, so must be the value of the estimator <math>X_{N}</math>. For the estimator to be ''unbiased'', the mean value of <math>X_{N}</math> must be true ensemble mean, <math>X</math>, i.e. | ||
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+ | <table width="100%"><tr><td> | ||
+ | :<math> | ||
+ | \lim_{N\rightarrow\infty} X_{N} = X | ||
</math> | </math> | ||
</td><td width="5%">(2)</td></tr></table> | </td><td width="5%">(2)</td></tr></table> |
Revision as of 10:02, 8 June 2006
Estimators for averaged quantities
Since there can never an infinite number of realizations from which ensemble averages (and probability densities) can be computed, it is essential to ask: How many realizations are enough? The answer to this question must be sought by looking at the statistical properties of estimators based on a finite number of realization. There are two questions which must be answered. The first one is:
- Is the expected value (or mean value) of the estimator equal to the true ensemble mean? Or in other words, is yje estimator unbiased?
The second question is
- Does the difference between the and that of the true mean decrease as the number of realizations increases? Or in other words, does the estimator converge in a statistical sense (or converge in probability). Figure 2.9 illustrates the problems which can arise.
Bias and convergence of estimators
A procedure for answering these questions will be illustrated by considerind a simple estimator for the mean, the arithmetic mean considered above, . For independent realizations where is finite, is given by:
| (2) |
Now, as we observed in our simple coin-flipping experiment, since the are random, so must be the value of the estimator . For the estimator to be unbiased, the mean value of must be true ensemble mean, , i.e.
| (2) |